At my father's funeral, Mum was pretty much true to form. She swore at me in the back of the limo all the way to the crematorium, stumbled out in a drunken haze and heckled the rabbi. Two weeks later she narrowly survived an overdose to stagger on alone and isolated in her detached house, alienating her friends one by one. When the neighbours noticed the milk hadn't been taken in, the police found her dead at the kitchen table, and the coroner eventually decided she'd died of natural causes. She'd been killed by a toxic combination of cigarettes, alcohol and grief.

By then our relationship was at its lowest ebb, which has made my sense of loss and guilt all the more profound. She was such a domineering character that I'd always felt her presence, no matter where I was. Since she died I've found myself dwelling more and more on her story, and the nagging awareness that somehow the final part of her tragedy could have been avoided if I'd been more able to stand up for her.

She'd been an heiress, a debutante, a concert-standard violinist – beautiful, talented and rich. It would be too simplistic to say that her later problems were those of a Holocaust survivor, because long before history made its mark on her things had already begun to go wrong.

Edith Szidonie Muller was one of two children of a family of entrepreneurs in Budapest, born at a pivotal moment in the history of that hedonistic city. Jews like my grandparents were so integral to its commercial and artistic life that some called it Judapest.

Amid this seethingly competitive society, my grandmother was desperate for the social cachet that would be hers with a prodigy in the family, so the young Edith was signed up for ballet classes. Family photo albums are packed with the evidence of success – en pointe at an improbably early age, dressed in Hungarian folk costume with hands on hips, or skating with one leg lifted in a graceful arabesque.

Then migraines struck, and the paediatrician insisted on putting a sudden end to the burgeoning ballet career. But my grandmother was not a woman to be so easily thwarted; she just changed tack, substituting intensive violin lessons for dance. Nor was the intensity of her ambition balanced by any penchant for outward expressions of affection; the atmosphere at home was austere and unwelcoming. If Edith missed a single practice session, she was beaten with her father's leather belt "until I was black and blue", she said.

Though she detested the enforced formality of the world she grew up in, it crept into her veins. After she died, the police found a note. "I couldn't say 'I love you' like they do in the American movies."

The Jews of Budapest have been described as sleepwalking towards a terrible fate. As Edith was growing up, draconian antisemitic legislation was shrinking their world, yet people continued to believe it would all pass. And it's true that even after the war broke out, homegrown fascists protected their Jews from the worst excesses of the Final Solution. Until the day in 1944 when the Germans marched into town and Adolf Eichmann masterminded the deportation of the Hungarian Jews to Auschwitz – the fastest in the history of the Third Reich.

It's impossible for us to imagine what it was like for a young teenager to see the people around her disappear one at a time, never to return. With Budapest besieged by the Russians, people took refuge from the bombardment in freezing, damp cellars, fighting over scraps of food. At night there was a further terror – fascist gangs roamed the city, pulling Jews out of the so-called safe houses, torturing them and shooting them into the river Danube. Twice my grandfather was rounded up, twice he survived because his 14-year-old daughter appealed to friends with connections in high places to intervene on his behalf.

As fighting broke out in the streets, the infrastructure collapsed into chaos and the much longed-for liberation, when it finally came, turned out to be a false dawn. The Soviet takeover brought its own brand of brutality. "I was raped by a Russian soldier," she once told me. Forty years on, she had to force the words out of her lips, then clamped them shut without elaborating and never spoke of it again. Maybe that's what prompted my grandparents to pack Edith off to a safer place. Armed with a set of forged papers bought on the black market, she headed for Paris with one suitcase and her violin. She was just 18.

More than a year later, an unlikely invitation brought her to Cardiff – a former employee of her father's was working there for a refugee-owned zip-making company. But soon the summer adventure across the Channel began to turn sour when her host, an older man she later described as ugly and charmless, tried to get his hands on her. She escaped into the arms of his boss's stepson, a striking young engineer from Berlin, and they married within two months. But the man who was to be my father had scars of his own; the child of a messy divorce, he'd been uprooted for ever from a country he never stopped thinking of as home. By the time they met, he'd already spent years in analysis with a colleague of Anna Freud. Looking back, I see they were just two young people clinging to the wreckage of their lives and hanging on to each other for support.

Yet the first five years of their marriage seems to have been a time of radiant happiness. Edith taught herself to cook and sew, running up a wardrobe full of dramatic New Look dresses. My father rose to chief engineer in the increasingly successful family business; Aero Zipp Fasteners now had a substantial share of the UK market. The couple socialised in a tight-knit group of young family and friends, mainly émigrés like themselves. And if the cultural life of the Welsh capital felt threadbare in those days, then they'd drive to London, see a play and have dinner. It seemed for a moment as if Edith had defied her past and remade her future. But when my brother and I arrived in 1955 and 1957, her demons returned, as if bringing us up reminded her of things she'd long since buried. From then on, the drink took hold, and stability disappeared through the door.

I'd come home from school to find her asleep in bed, reeking of booze. Often I'd finish cooking for guests because she was no longer capable. At nights, when my father was out at the bridge club, she would phone friends, relations, even business colleagues of my father for rambling, alcohol-fuelled conversations. Social situations were obstacle courses we had to struggle over, never knowing whether she would be fabulous fun or fall over drunk.

My parents' relationship continued to be a mass of contradictions. On one level they adored each other. They were a team, their ultimate delight eating and drinking their way across Europe for business and pleasure. She was the map-reader, armed with a library of travel guides, phrase books and newspaper cuttings, constantly on the lookout for a new hotel or restaurant to add to her itinerary.

But back home, the fun could soon evaporate. After the family business was sold in the 1970s, my father had set up his own engineering company, which he was extremely proud of but which enjoyed mixed fortunes. Edith struggled to deal with the financial unpredictability, and this exacerbated her boozing, which put a constant strain on my father. When they moved to the Midlands in the 1980s, I got the impression of increasing social isolation. She'd always been emotionally dependent on him, but now that got a good deal worse. At times I caught a whiff of desperation, and after he died of cancer in 1991 I discovered he'd covered up several attempts she'd made at suicide.

Though Edith had spent years in therapy, at home her problems were often brushed under the carpet. On good days she could be funny and entertaining. When I was growing up, she entertained lavishly and frequently, loving nothing more than a house full of people with big appetites. Well read and clever, she breezed her way through the Telegraph crossword every day in what was, after all, her fourth language. When I was old enough for shopping trips to London, she'd let me have anything I wanted, taking me to cosy bistros for delicious lunches that seemed to my teenage self the acme of metropolitan sophistication. In the summer she used to summon all the girls in the street to handstand classes on our front lawn, using me as a demonstrator. She'd stand there, cigarette-holder clamped between yellowing teeth, barking out instructions.

But that's not what people remember. After she died, the verdict of old friends was surprisingly unanimous, "Ach, you know, she meant well," they said, forgetting they'd indulged themselves at her table and enjoyed her company. Faint praise seemed to airbrush away her huge range of talents and skills. "She meant well" was a way of saying that the drunken, domineering Edith always proved more memorable than the talented, clever one or the witty raconteur.

I'm as guilty as any of them. Fabulous she may have been, but living with her was like being on an emotional pendulum, and I left home exhausted as soon as I could, heading to Europe for a gap year and then university. I never really went back. I kept my feelings to myself, giving my parents a carefully edited version of my life. I never actually cut them off, just didn't let them in. It was the only way I knew how to survive. My brother, gentler and more forgiving than me, returned from boarding school and stayed close to home in a way I'd have found impossible.

After I married, I used my own family as a fortress to hide in. By now my mother was a widow living alone, but I felt little empathy for her loneliness or her grief. Each time we visited, she'd be drunk when we arrived, which would simply confirm my entrenched view of her as a hopeless headcase.

For years I was angry with Edith for making me grow up with her problems, and resentful at both my parents for behaving as though the drunkenness and emotional vagaries were simply not happening. But bringing up my own family has made me realise how easy it is to be critical. I've no more succeeded in protecting my three children from being dragged into my own emotional turmoil than she did, and I certainly haven't got the excuses that she had. My fury blinded me to her incredible strengths as a parent – the freedom she allowed me, the generosity with which she supplied my needs, her willingness to listen if I'd ever been minded to talk to her.

Despite our frigid relationship at the end, she was hugely proud of my independence, emotional health (everything's relative) and, most of all, my professional achievements. When I got my first job in television as a researcher, both my parents thought it was some kind of miracle. To them, the BBC appeared to be an impenetrable bastion of the British establishment, and each time I got promoted, they thought I'd broken through another layer of informal Jewish quota.

Finally, more than a decade after her death, I realised that my mother was no worse a parent than many of the ones who said so patronisingly that Edith "meant well". They may have been better at covering things up than she was, but looking round at my generation, the children of refugees and survivors, it's striking how many of us have been scarred in some way by what our parents went through. My mother was all too well aware of that, and the truth is that she did her best to set me free, regardless of the cost to herself.